Background error modelling: climatological flow-dependence
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1 Background error modelling: climatological flow-dependence Yann MICHEL NCAR/MMM/B Meeting 16 th April Introduction 2 A new estimate of lengthscales 3 Climatological flow-dependence Yann MICHEL B modelling: climatological flow-dependence 1
2 Background error modelling: climatological flow-dependence Spectral formulations of the B matrix allow: three-dimensional grid point background error σ b ; good representation the average spatial structure of analysis increments. But they are restricted to homogeneous isotropic horizontal analysis increments (for CV); homogeneous structure of vertical correlation (for CV). Gridpoint formulations They allow to relax those assumptions: EOF decomposition can be local; recursive filters can be applied with varying lengthscales. Yann MICHEL B modelling: climatological flow-dependence 2
3 Background error modelling: climatological flow-dependence However currently gen_be/wrf-var do not make benefit of this! We end up with the worse covariances of both worlds! homogeneous and isotropic structure of horizontal analysis increments (for CV), that have a bad power spectrum (Gaussian assumption) homogeneous structure of vertical correlation (for CV) homogeneous σ b (they are the sqrt of EOF eigenvalues) Moreover, the horizontal analysis increments looks not so good (c.f. PSOT results of gen_be_stage4_regional without tuning of lengthscales). Yann MICHEL B modelling: climatological flow-dependence 3
4 A new estimate of lengthscales The Wu, Purser and Parrish (2002) formula The correlation lengthscale can be estimated through the ration of the variance of the field to the variance of its Laplacian: L = 8 V «1/4 ψ V ζ gen_be_stage4_regional_from_variances * for each member (date or ensemble) + for each vertical level or EOF mode - perform spatial laplacian computation with fft - accumulate variance for projected field - accumulate variance for projected Laplacian field + end * end * Compute lengthscales Yann MICHEL B modelling: climatological flow-dependence 5
5 A new estimate of lengthscales: results for CONUS 200 We obtain local lengthscales which noise seems in agreement with Pannekoucke et al (2008) and choose to use the median over the domain. Figure: Comparison of raw lengthscales obtained over the CONUS domain (200 km resolution) for the fit of spatial correlation (gen_be_stage4_regional) and the new estimate from the variances (gen_be_stage4_regional_from_variances) Yann MICHEL B modelling: climatological flow-dependence 6
6 A new estimate of lengthscales: results for AMPSRT 45 Comparison Smoother results over the EOF mode (more robust). Shorter lengthscales for ψ, χ u and larger for t u, rh. Figure: Same comparison over the AMPSRT domain (45 km resolution) Yann MICHEL B modelling: climatological flow-dependence 7
7 Single observation experiments (PSOT) Figure: PSOT over the AMPSRT domain (45 km resolution) for a u observation Yann MICHEL B modelling: climatological flow-dependence 8
8 Single observation experiments (PSOT) Figure: PSOT over the AMPSRT domain (45 km resolution) for a u observation Yann MICHEL B modelling: climatological flow-dependence 8
9 Single observation experiments (PSOT) Figure: PSOT over the AMPSRT domain (45 km resolution) for a t observation Yann MICHEL B modelling: climatological flow-dependence 9
10 Single observation experiments (PSOT) Figure: PSOT over the AMPSRT domain (45 km resolution) for a t observation Yann MICHEL B modelling: climatological flow-dependence 9
11 Single observation experiments (PSOT) Figure: PSOT over the AMPSRT domain (45 km resolution) for a rh observation Yann MICHEL B modelling: climatological flow-dependence 10
12 Modelling the climatological flow-dependence Currently, the bin_type in gen_be is an on-off way of specifying B. It allows j-dependence of Regression coefficients (stage 2, U p ) Eigenvalues/vectors of vertical covariance (stage 3, U v ) Lengthscales (stage 4, U h ) But when the grid is not j/latitude (as for AMPSRT), we are restricted to homogeneity (bin_type 5) x = U p U v U h v (1) Yann MICHEL B modelling: climatological flow-dependence 11
13 Varying background error variances gen_be_stage4_regional_from_variances provides gridpoint variances of fields on EOF modes that could be used to add climatological dependence of background error variances. Figure: Variance for ψ and χ u on EOF 1 Yann MICHEL B modelling: climatological flow-dependence 12
14 Varying background error lengthscales gen_be_stage4_regional_from_variances provides gridpoint lengthscales of fields on EOF modes Figure: Lengthscale for ψ and χu on EOF 1 Yann MICHEL B modelling: climatological flow-dependence 13
15 Varying background error lengthscales: recursive filters Recursive filters can deal with smoothly varying lengthscales, but the amplitude of the filter has to be reconsidered (Purser et.al, 2003). Basic equation of recursive filter of smoothing parameter α can easily be made grid-dependent. A i = αa i 1 + (1 α)a i Figure: A varying lengthscale (km) Impulse Figure: Impulse response of the 6-order recursive filter. Yann MICHEL B modelling: climatological flow-dependence 14
16 Varying background error lengthscales in WRFVAR An academic test where the background error lengthscales are increasing by a factor of 2 from West to East. Still some work to go to achieve proper normalization of the amplitude of the recursive filters. Yann MICHEL B modelling: climatological flow-dependence 15
17 Conclusion Climatological flow-dependence New way of computing lengthscales, more efficient, looks better. On the way of specifying varying background error variances On the way of specifying varying background error lengthscales With bin_type=0, one can have varying regression coefficients (not shown) Filtering Variances, Lengthscales and Regression coefficients may need to be locally averaged (spatially and or temporally averaged). The filter could be adaptive to the noise level and structure (Berre, Raynaud, Pannekoucke). Flow-dependence of the day This improvements could be used with the hybrid framework, but still a lot of work before. Yann MICHEL B modelling: climatological flow-dependence 16
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